**1. Introduction**

Sir William Henry Bragg and his son William Lawrence Bragg were the pioneers of crystallography (Bragg, W. H., 1912, 1913a, 1915a, 1915b; Bragg, W. L., 1920). In 1913, they published several articles, notably *The reflection of X-rays by crystals* (Bragg, W. H., 1913b) and *The structure of diamond* (Bragg, W. H. & Bragg, W. L., 1913) were they wrote: "We have applied the new methods of investigation involving the use of X-rays to the case of the diamond, and have arrived at a result which seems of considerable interest. The structure is extremely simple". Two years later, they were jointly awarded the Nobel Prize in Physics for their works in the analysis of crystal structure by means of X-rays.

A century after the first crystallographic experiment, new computing facilities, modern technologies and new diffraction sources (synchrotron, neutron sources…) offer a large range of possibilities and opportunities for crystallographers to probe matter. Crystallography appears nowadays as a new science.

Performing structural analyses at ambient conditions or at low temperature (i.e. above 100 K using nitrogen jet-stream) is very common and popular in laboratories to obtain the structure of powder and single-crystal materials. To understand the mechanisms governing the behaviour of materials it is essential to well know the close relations between the structure and the physical and chemical properties. First, X-ray diffraction is a unique tool to obtain routinely a detailed description of atomic structure and thermal vibrations by analysing the diffracted intensities *Ihkl* of the crystallographic *hkl* reflections:

$$I\_{hkl} = \mathbf{S} \mathbf{C}\_{hkl} \cdot \left| F\_{hkl} \right|^2 \tag{1}$$

$$F\_{hkl} = \sum\_{j=1}^{n} f\_j.T\_j. \exp(2i\pi(h.x\_j + k.y\_j + l.z\_j))\tag{2}$$

© 2012 Vincent, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Vincent, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 40 Recent Advances in Crystallography

where S is a scale factor, Chkl is an experimental corrections term (including absorption, extinction, Lorentz-polarization correction…), Fhkl is the structure factor of a *hkl* reflection which depends of fj the form factor of the atom j with (xj, yj, zj) coordinates in the cell and of Tj the Debye-Waller factor given in the case of isotropic harmonic vibrations by:

$$T\_j = \exp\left(-B\_j \frac{\sin^2 \theta}{\lambda^2}\right) = \exp\left(-8\pi^2 \left\langle \mathcal{U}\_j^2 \right\rangle \frac{\sin^2 \theta}{\lambda^2}\right) \tag{3}$$

where <sup>2</sup> *Uj* is the mean square atomic displacement of the atom j.

If one wants to increase the data quality, the best choice, without consideration about crystal quality or particular experimental conditions, is to perform a low temperature measurement, usually at 110 K, to reduce the atomic displacement parameters which affect the value of the atomic structure factor. In those experimental conditions, if high resolution X-ray diffraction measurement is performed up to large momentum transfers (sinθ/λ > 1 Å-1) the electron density distribution of a molecule can be determined thanks to the Hansen-Coppens pseudo-atomic multipolar expansion (Hansen & Coppens, 1978):

$$\rho(\vec{r}) = \rho\_{\rm core}(r) + P\_{\rm vol} \kappa^3 \rho\_{\rm vol}(\kappa r) + \sum\_{l} \kappa^{\star 3} R\_{nl}(\kappa^{\star} r) \sum\_{m=0}^{l} P\_{lm \pm} y\_{lm \pm}(\theta, \rho) \tag{4}$$

where *ρcore(r)* and *ρval(r)* are spherically averaged core and valence electron densities calculated from Clementi Hartree-Fock wave functions for ground state isolated atoms (Clementi & Roetti, 1974). *κ* and *κ'* are contraction-expansion parameters and *Pval* is the atomic valence shell electron population. The deformation of the valence electron shell is projected on real spherical harmonics *ylm±(θ,φ)* times Slater-type radial functions *Rnl(r)*. *Plm±* are the multipole population parameters.

In the later case, the accuracy of the results is highly dependant of the experimental conditions (Destro et al., 2004; Zhurov et al., 2008) and particularly of the crystal quality and the measurement temperature which must be chosen to reduce at the maximum the thermal vibrations of the atoms. These *usual experiments* at static low temperature give basic structural properties, and their variations as a function of temperature can reveal particular behaviour of matter as phase transition for example with changes in structural, electronic, optical and/or magnetic properties. But, in a general consideration, performing crystallographic measurements under external perturbations is of prime importance. Nowadays, experiments at different temperatures or hydrostatic pressures can be done almost routinely and exotic sample environments are more and more used to explore materials properties. If we just consider for the moment experiments involving temperature or pressure variations, a Web of Science search for "high pressure", "low temperature" and "high temperature" in the field of X-ray and neutron diffraction between the years 2005 and 2012 gave about 3500 experiments performed in 2005 but nearly 5000 in 2011 (figure 1). Two comments can be done observing figure 1. First, a large amount of the extreme conditions (temperature and pressure) experiments are carried out using X-ray diffraction compared to neutron diffraction. This difference is of course due to the large number of X-ray diffractometers available in laboratories, which are more and more outstanding in terms of source power and detector efficacy, combined with numerous sample environments especially designed for laboratory equipments. But this is also due to the building of several 3rd generation of synchrotron sources which offer the possibility to conduct very quick measurements on a very small quantity of matter, allowing the measurement of materials under non ambient conditions, particularly in the domains of chemistry and biology sciences (neutron diffraction experiment demands more matter quantity, in general at least some mm3). The second observed aspect on figure 1 concerns the evolution of the cumulative number (neutron and X-ray) of publications during the considered period: about 300 supplementary published papers per year appear between 2005 and 2009, but in the years 2009-2011, a distinct decrease of this number is noticed. This effect is directly related to the number of synchrotron radiation facilities over the world. One can count about 69 particle accelerators and accelerator laboratories in 2005 and about 76 in 2012 (including about 40 synchrotrons in 2011), with most of the new facilities operative in 2006-2008.

40 Recent Advances in Crystallography

are the multipole population parameters.

where S is a scale factor, Chkl is an experimental corrections term (including absorption, extinction, Lorentz-polarization correction…), Fhkl is the structure factor of a *hkl* reflection which depends of fj the form factor of the atom j with (xj, yj, zj) coordinates in the cell and of

> 

If one wants to increase the data quality, the best choice, without consideration about crystal quality or particular experimental conditions, is to perform a low temperature measurement, usually at 110 K, to reduce the atomic displacement parameters which affect the value of the atomic structure factor. In those experimental conditions, if high resolution X-ray diffraction measurement is performed up to large momentum transfers (sinθ/λ > 1 Å-1) the electron density distribution of a molecule can be determined thanks to the

2 2

2 2 sin sin

2 2

(3)

 

0

 

*l*

Tj the Debye-Waller factor given in the case of isotropic harmonic vibrations by:

where <sup>2</sup> *Uj* is the mean square atomic displacement of the atom j.

 

exp exp 8 *j j <sup>j</sup> T B U* 

Hansen-Coppens pseudo-atomic multipolar expansion (Hansen & Coppens, 1978):

3 3

*r r P r R r Py*

() () ( ) ' ( ') ( , )

*core val val nl lm lm*

where *ρcore(r)* and *ρval(r)* are spherically averaged core and valence electron densities calculated from Clementi Hartree-Fock wave functions for ground state isolated atoms (Clementi & Roetti, 1974). *κ* and *κ'* are contraction-expansion parameters and *Pval* is the atomic valence shell electron population. The deformation of the valence electron shell is projected on real spherical harmonics *ylm±(θ,φ)* times Slater-type radial functions *Rnl(r)*. *Plm±*

In the later case, the accuracy of the results is highly dependant of the experimental conditions (Destro et al., 2004; Zhurov et al., 2008) and particularly of the crystal quality and the measurement temperature which must be chosen to reduce at the maximum the thermal vibrations of the atoms. These *usual experiments* at static low temperature give basic structural properties, and their variations as a function of temperature can reveal particular behaviour of matter as phase transition for example with changes in structural, electronic, optical and/or magnetic properties. But, in a general consideration, performing crystallographic measurements under external perturbations is of prime importance. Nowadays, experiments at different temperatures or hydrostatic pressures can be done almost routinely and exotic sample environments are more and more used to explore materials properties. If we just consider for the moment experiments involving temperature or pressure variations, a Web of Science search for "high pressure", "low temperature" and "high temperature" in the field of X-ray and neutron diffraction between the years 2005 and 2012 gave about 3500 experiments performed in 2005 but nearly 5000 in 2011 (figure 1). Two comments can be done observing figure 1. First, a large amount of the extreme conditions (temperature and pressure) experiments are carried out using X-ray diffraction compared to neutron diffraction. This difference is of course due to the

 

*l m*

 

(4)

As said before, crystallography under extreme conditions is now more and more used and plays a key role as it offers helpful understanding of the physical, chemical and mechanical properties of the solid state. The term *extreme conditions* was first used to define non ambient thermodynamical conditions of pressure or temperature. It is also now employed when outof-equilibrium conditions are applied such as light irradiation, external magnetic or electric fields, specific chemical environments (e.g. under liquid or gas flux…) or applied strain. In situ measurements can also be considered as extreme conditions, for examples in the cases of time-resolved experiments (picosecond diffraction…) or chemical kinetic reactions (dehydrogenation reaction, diffusion process, decomposition pathway…). The present challenge is to combine two or more extreme conditions to explore new states of matter and new material properties, taking advantage of last generations of high brilliance sources (synchrotron and neutron sources) (figure 2).

**Figure 1.** Publication number related to neutron or X-ray diffraction experiments under extreme conditions of temperature and pressure during the period 2005-2011 (Web of Science, March 2012).

#### 42 Recent Advances in Crystallography

When extreme conditions are applied to a material, various changes occur. They may involve sample state variations (gaseous, liquid and solid phases), phase transitions (magnetic and structural), electronic structure modifications (the chemical bonds can change from covalent to ionic or metallic) and atomic bond lengths variations (which induce variations of the atomic vibrations, of the coordination numbers, of the diffusion processes…). A large number of the material properties are modified under extreme conditions, leading sometimes to metastable states, which open new investigation opportunities for the crystallographer, but also for all scientist who wants to go deeper in the understanding of the structure-properties relationships in order to design future materials.

**Figure 2.** The neutron source of the Institut Laue-Langevin (ILL) and the European Synchrotron Radiation Facility (ESRF) in Grenoble, France (credit: V. Legrand).

This chapter is reviewing different aspects of the use of crystallography under extreme conditions to investigate the nature, the mechanisms and the dynamics of out-ofequilibrium phase transitions as well as transformations driven by external applied conditions. The part 2. is focus on the advantages of using Large scale facilities around the world and what are now the extreme sample environments available to explore unknown properties of materials. Finally, an overview of perspectives and future developments in diffraction instruments and sample environments are presented in part 3. showing that modern crystallography is in perpetual evolution and that the present century is certainly the one of *crystallography under extreme conditions*.
